Gauss's Law

Example: Uniform Spherical
Charge

Consider a uniform spherical distribution of charge.
This must be charge held in place in an insulator. Charge on a
conductor would be free to move and would end up on the surface.
This charge density is uniform throughout the sphere.

Charge Q is uniformly distributed throughout a sphere of
radius a. Find the electric field at a radius r.

First consider r > a; that is, find the electric field at a
point outside the sphere.

Just as before (for the point charge), we start with Gauss's
Law

Just as for the point charge, we find

and we know

which means

E = k Q / r2

That is, the electric field outside the sphere is
exactly the same as if there were only a point charge Q.

That is, the electric field inside the sphere of uniform
charge is zero at the center and increases
linearlywith radius r:

Of course, the two expressions for the electric field match --
have the same value -- at the surface of the sphere, for r = a.

Using Gauss's Law here made the "calculation" almost easy. A
more direct application of Coulomb's Law -- with a detailed
integration over the volume and a careful consideration of
the vector nature of Coulomb's Law would have been far
more difficult.